Powers and Roots of Negative Numbers

1. Section 8-7
Powers and Roots of Negative Numbers

2. Warm-up
Complete the In-Class Activity on p. 510

3. n
(−x )

4. n
(−x )
Positive when n is even

5. n
(−x )
Positive when n is even
Negative when n is odd

6. n
(−x )
Positive when n is even
Negative when n is odd
n can be any real number:
positive, negative or zero

7. Example 1
Simplify.
6
4 −6 (−5)
a. (−3) • (−3) b. 3
(−5)

8. Example 1
Simplify.
6
4 −6 (−5)
a. (−3) • (−3) b. 3
(−5)
−2
= (−3)

9. Example 1
Simplify.
6
4 −6 (−5)
a. (−3) • (−3) b. 3
(−5)
−2
= (−3)
1
= 2
(−3)

10. Example 1
Simplify.
6
4 −6 (−5)
a. (−3) • (−3) b. 3
(−5)
−2
= (−3)
1
= 2
(−3)
1
=
9

11. Example 1
Simplify.
6
4 −6 (−5)
a. (−3) • (−3) b. 3
(−5)
−2
= (−3)
3
= (−5)
1
= 2
(−3)
1
=
9

12. Example 1
Simplify.
6
4 −6 (−5)
a. (−3) • (−3) b. 3
(−5)
−2
= (−3)
3
= (−5)
1
= 2
(−3) = −125
1
=
9

13. x is real, −x is imaginary, x ≥ 0

14. x is real, −x is imaginary, x ≥ 0
When x and y are negative, x • y ≠ xy

15. x is real, −x is imaginary, x ≥ 0
When x and y are negative, x • y ≠ xy
−4 • −9 ≠ 36

16. x is real, −x is imaginary, x ≥ 0
When x and y are negative, x • y ≠ xy
−4 • −9 ≠ 36
Why?

17. x is real, −x is imaginary, x ≥ 0
When x and y are negative, x • y ≠ xy
−4 • −9 ≠ 36
Why?
−4 = 2i

18. x is real, −x is imaginary, x ≥ 0
When x and y are negative, x • y ≠ xy
−4 • −9 ≠ 36
Why?
−4 = 2i −9 = 3i

19. x is real, −x is imaginary, x ≥ 0
When x and y are negative, x • y ≠ xy
−4 • −9 ≠ 36
Why?
−4 = 2i −9 = 3i 36 = 6

20. x is real, −x is imaginary, x ≥ 0
When x and y are negative, x • y ≠ xy
−4 • −9 ≠ 36
Why?
−4 = 2i −9 = 3i 36 = 6
2
2i • 3i = 6i

21. x is real, −x is imaginary, x ≥ 0
When x and y are negative, x • y ≠ xy
−4 • −9 ≠ 36
Why?
−4 = 2i −9 = 3i 36 = 6
2
2i • 3i = 6i = −6

22. x is real, −x is imaginary, x ≥ 0
When x and y are negative, x • y ≠ xy
−4 • −9 ≠ 36
Why?
−4 = 2i −9 = 3i 36 = 6
2
2i • 3i = 6i = −6 ≠ 6

23. 3 6
When x < 0, x ≠ x . Why?

24. 3 6
When x < 0, x ≠ x . Why?
Let x = -3

25. 3 6
When x < 0, x ≠ x . Why?
Let x = -3
3 6
(−3) ≠ (−3)

26. 3 6
When x < 0, x ≠ x . Why?
Let x = -3
3 6
(−3) ≠ (−3)
−27 ≠ 729

27. 3 6
When x < 0, x ≠ x . Why?
Let x = -3
3 6
(−3) ≠ (−3)
−27 ≠ 729
−27 ≠ 27

28. 3 6
When x < 0, x ≠ x . Why?
Let x = -3
3 6
(−3) ≠ (−3)
−27 ≠ 729
−27 ≠ 27
(You will see noninteger powers of negative numbers in FST)

29. We can take nth roots of negative numbers

30. We can take nth roots of negative numbers
When x is negative and n is an odd integer > 2,
the real nth root of x is
n
x

31. Example 2
Simplify.
7
−16384

32. Example 2
Simplify.
7
−16384
= −4

33. Theorem
n
When x and n y are defined and real numbers,
n
then n xy is also defined and n xy = x • n y

34. Theorem
n
When x and n y are defined and real numbers,
n
then n xy is also defined and n xy = x • n y
We cannot do this with even roots of negative numbers

35. Example 3
Simplify.
5
−3645

36. Example 3
Simplify.
5
−3645
5 5
= −243 • 15

37. Example 3
Simplify.
5
−3645
5 5
= −243 • 15
5
= −3 15

38. Homework
p. 515 #1-26
“Wanting to be someone you’re not is a waste of the
person you are.” - Kurt Cobain